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A barnyard is to be fenced is as indicated in Figure $21 .$ If 3600 feet of fencing is to be used in its construction determine the dimension $x$ and $y$ that will maximize its total area?

$$x=360 \mathrm{ft}, y=900 \mathrm{ft}$$

Algebra

Chapter 1

Functions and their Applications

Section 4

Quadratic Functions - Parabolas

Functions

Harvey Mudd College

Baylor University

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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A barnyard is to be fenced…

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All right guys, today, we're gonna be looking at a very simple optimization question. Now, optimization questions, basically optimizing or finding the best possible value given um Only a certain amount of information. So in this case what we're optimizing is maximizing the total area of this. Um what is it barnyard given? The 3600? Right, so this is like the dimensions that were given in the question, I just roughly sketched out, we know that we have five X. Is that we need to cover right in our coronary defense thing? Have 5x. is right? Um now the length of the Barnard right is depicted by why? So we have two wives that we also need a coverage five. Right? So this is the total amount of fencing that we must do in department and this total fencing will all equal to over here. 3600, 30 600. Um total feet of this is our based formula here. Um Now what we're going to have to do is simplify and kind of solve for one of these variables. So why it seems a little the east in this case one of the y equals uh y equals 3. 600. Right, nice. Uh cheap clothes, 1500 were to add Which is 2.5. Usually we're not, you know, supposed to use decimals. Um but because it is a very simple decimal, just 2.5, there's no extending numbers um like 3 63 or 60 it's fine to this. Uh Right, so now we have solved for why what we're gonna do now is find the area of this apartment, so we know that area. Okay, so this is a very rectangular shape is just excellent. Why? All right, we already know why over here, so we just thanks and and thank you. Uh the scroll down. Yeah. Um now what we're gonna do is simplify it. So we're just gonna send the exit out here. We have 1800. Right? Right now this here is our area function, but um something that we can notice here is that this function is actually a quadratic right? It's got an expert. So if we were to graph it out um using either a graphing calculator or any kind of graphing software, we're going to find that this would be a terrible right? Um Let me actually rewrite it. So mm right, this here is a parabola. Um and because it's a negative, we know that if we were to draw this out it would look something, it would be a downward facing to maximize the area of this travel. We have to find it's the vertex. Alright, What we're trying to look for is the vertex right? And first find the X. Value of the vertex. We're gonna use a formula here. So the X value of vortex. So I usually write be in a subscript X. So the X value of vertex is simply minus B. Right? In this case this here is beat and right. So the B value right? We're going to go negative 1800 right over to times this is negative to the point. Yes. Um to simplify this quickly We got negative 1800 yes, both five. Which simplifies to which number 18? Ah 360. 306. What does this tell us? This tells us the X values. So if you go all the way up to the top this right here. Mhm. Right. The X value will be Mhm. one of those exes is sorry Um one of those excesses. 360s. The what? Okay. Now to find the why you guys remember we simplified for why over here this is the Y. Equation. Right? So now to find out why we're just gonna sub 360. I'm right here into this. Why question? So uh we're gonna have Okay. Mhm. Double check. It was 1800 2.5 X. And now we're going to actually just sub index here. So we got 300. Beautiful. Now just going to type this into my calculator. So we have 1800 900 miles is clear in that. There you go. So we have now 900 ft for the white. That means that this over here. The wise are each 900 ft and left. So that is um in essence our vertex too. So if you were to actually grasp it out like I did over here, right? Yeah, vertex would be in a different color. The vertex would be over here and then it would be a downward facing threat. So this vertex over here really shows us the maximizing potential for the area. Right? So to maximize um the total area of the barnyard, the width of the X. Value would have to be three uncertainly Actually have people 360 ft. And why would have to call that? In essence, would maximize the total um total area of? All? Right, perfect. We'll see in the next one. Mhm.

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